Estimates of the higher-order QCD corrections: Theory and Applications

We consider the further development of the formalism of the estimates of higher-order perturbative corrections in the Euclidean region, which is based on the application of the scheme-invariant methods, namely the principle of minimal sensitivity and the effective charges approach. We present the estimates of the order $O(\alpha^{4}_{s})$ QCD corrections to the Euclidean quantities: the $e^+e^-$-annihilation $D$-function and the deep inelastic scattering sum rules, namely the non-polarized and polarized Bjorken sum rules and to the Gross--Llewellyn Smith sum rule. The results for the $D$-function are further applied to estimate the $O(\alpha_s^4)$ QCD corrections to the Minkowskian quantities $R(s) = \sigma_{tot} (e^{+}e^{-} \to {\rm hadrons}) / \sigma (e^{+}e^{-} \to \mu^{+} \mu^{-})$ and $R_{\tau} = \Gamma (\tau \to \nu_{\tau} + {\rm hadrons}) / \Gamma (\tau \to \nu_{\tau} \overline{\nu}_{e} e)$. The problem of the fixation of the uncertainties due to the $O(\alpha_s^5)$ corrections to the considered quantities is also discussed.Comment: revised version and improved version of CERN.TH-7400/94, LATEX 10 pages, six-loop estimates for R(s) in Table 2 are revised, thanks to J. Ellis for pointing numerical shortcomings (general formulae are non-affected). Details of derivations of six-loop estimates for R_tau are presente

The renormalization group inspired approaches and estimates of the tenth-order corrections to the muon anomaly in QED

We present the estimates of the five-loop QED corrections to the muon anomaly using the scheme-invariant approaches and demonstrate that they are in good agreement with the results of exact calculations of the corresponding tenth-order diagrams supplemented by the additional guess about the values of the non-calculated contributions.Comment: LATEX 15 pages, figures available upon request; preprint CERN-TH.7518/9